Estimating jump-diffusions using closed-form likelihood expansions. (English) Zbl 1443.62361
Summary: The indispensable role of likelihood expansions in financial econometrics for continuous-time models has been established since the ground-breaking work of Y. Aït-Sahalia [in: Quantitative analysis in financial markets. Collected papers of the New York University Mathematical Finance Seminar. Vol. II. Singapore: World Scientific. 1–34 (2001; Zbl 1013.91505); Econometrica 70, No. 1, 223–262 (2002; Zbl 1104.62323); Ann. Stat. 36, No. 2, 906–937 (2008; Zbl 1246.62180)]. Jump-diffusions play an important role in modeling a variety of economic and financial variables. As a significant generalization of C. Li [Ann. Stat. 41, No. 3, 1350–1380 (2013; Zbl 1273.62196)], we propose a new closed-form expansion for transition density of Poisson-driven jump-diffusion models and its application in maximum-likelihood estimation based on discretely sampled data. Technically speaking, our expansion is obtained by perturbing paths of the underlying model; correction terms can be calculated explicitly using any symbolic software. Numerical examples and Monte Carlo evidence for illustrating the performance of density expansion and the resulting approximate MLE are provided in order to demonstrate the practical applicability of the method. Using the theoretical results in [M. Hayashi and Y. Ishikawa, Math. Nachr. 285, No. 5–6, 619–658 (2012; Zbl 1252.60051)], some convergence properties related to the density expansion and the approximate MLE method can be justified under some standard sufficient (but not necessary) conditions.
MSC:
| 62P05 | Applications of statistics to actuarial sciences and financial mathematics |
| 62M05 | Markov processes: estimation; hidden Markov models |
| 60J60 | Diffusion processes |
| 62F12 | Asymptotic properties of parametric estimators |
| 91G20 | Derivative securities (option pricing, hedging, etc.) |
Keywords:
maximum-likelihood estimation; jump-diffusion; discrete observations; transition density; expansionReferences:
| [1] | Abate, J.; Whitt, W., The Fourier-series method for inverting transforms of probability distributions, Queueing Syst. Theory Appl., 10, 5-87 (1992) · Zbl 0749.60013 |
| [2] | Aït-Sahalia, Y., Transition densities for interest rate and other nonlinear diffusions, J. Finance, 54, 1361-1395 (1999) |
| [3] | Aït-Sahalia, Y., Maximum-likelihood estimation of discretely-sampled diffusions: a closed-form approximation approach, Econometrica, 70, 223-262 (2002) · Zbl 1104.62323 |
| [4] | Aït-Sahalia, Y., Telling from discrete data whether the underlying continuous-time model is a diffusion, J. Finance, 57, 2075-2112 (2002) |
| [5] | Aït-Sahalia, Y., Disentangling diffusion from jumps, J. Financ. Econ., 74, 487-528 (2004) |
| [6] | Aït-Sahalia, Y., Closed-form likelihood expansions for multivariate diffusions, Ann. Statist., 36, 906-937 (2008) · Zbl 1246.62180 |
| [7] | Aït-Sahalia, Y.; Cacho-Diaz, J.; Hurd, T., Portfolio choice with a jumps: A closed form solution, Ann. Appl. Probab., 19, 556-584 (2009) · Zbl 1170.91364 |
| [8] | Aït-Sahalia, Y.; Cacho-Diaz, J.; Laeven, R. J., Modeling financial contagion using mutually exciting jump processes, J. Financ. Econ., 117, 585-606 (2015) |
| [9] | Aït-Sahalia, Y.; Hurd, T., Portfolio choice in markets with contagion, J. Financ. Econ., 14, 1-28 (2015) |
| [10] | Aït-Sahalia, Y.; Kimmel, R., Maximum likelihood estimation of stochastic volatility models, J. Financ. Econ., 83, 413-452 (2007) |
| [11] | Aït-Sahalia, Y.; Kimmel, R., Estimating affine multifactor term structure models using closed-form likelihood expansions, J. Financ. Econ., 98, 113-144 (2010) |
| [12] | Aït-Sahalia, Y.; Mykland, P., The effects of random and discrete sampling when estimating continuous-time diffusions, Econometrica, 71, 483-549 (2003) · Zbl 1142.60381 |
| [13] | Aït-Sahalia, Y.; Mykland, P., Estimators of diffusions with randomly spaced discrete observations: a general theory, Ann. Statist., 32, 2186-2222 (2004) · Zbl 1062.62155 |
| [14] | Aït-Sahalia, Y.; Yu, J., Saddlepoint approximations for continuous-time markov processes, J. Econometrics, 134, 507-551 (2006) · Zbl 1418.62286 |
| [15] | Andersen, T.; Benzoni, L.; Lund, J., An empirical investigation of continuous-time equity return models, J. Finance, 57, 1239-1284 (2002) |
| [16] | Bakshi, G.; Ju, N.; Ou-Yang, H., Estimation of continuous-time models with an application to equity volatility dynamics, J. Financ. Econ., 82, 227-249 (2006) |
| [17] | Bates, D. S., Jumps and stochastic volatility: Exchange rate processes implicit in Deutsche mark options, Rev. Financ. Stud., 9, 69-107 (1996) |
| [18] | Bates, D. S., Post-’87 crash fears in the S&P 500 futures option market, J. Econometrics, 94, 181-238 (2000) · Zbl 0942.62118 |
| [19] | Black, F.; Scholes, M. S., The pricing of options and corporate liabilities, J. Polit. Econ., 81, 637-654 (1973) · Zbl 1092.91524 |
| [20] | Broadie, M.; Chernov, M.; Johannes, M., Model specification and risk premia: Evidence from futures options, J. Finance, 62, 1453-1490 (2007) |
| [21] | Chang, J.; Chen, S., On the approximate maximum likelihood estimation for diffusion processes, Ann. Statist., 39, 2820-2851 (2011) · Zbl 1246.62181 |
| [22] | Cheridito, P.; Filipović, D.; Kimmel, R. L., Market price of risk specifcations for affine models: Theory and evidence, J. Financ. Econ., 83, 123-170 (2007) |
| [23] | Chernov, M.; Gallant, R.; Ghysels, E.; Tauchen, G., Alternative models for stock price dynamics, J. Econometrics, 116, 225-257 (2003) · Zbl 1043.62087 |
| [24] | Choi, S., Closed-form likelihood expansions for multivariate time-inhomogeneous diffusions, J. Econometrics, 174, 45-65 (2013) · Zbl 1283.60106 |
| [26] | Choi, S., Explicit form of approximate transition probability density functions of diffusion processes, J. Econometrics, 187, 57-73 (2015) · Zbl 1337.62202 |
| [27] | Cinlar, E.; Jacod, J., (Representation of semimartingale Markov processes in terms of Wiener processes and Poisson random measures. Representation of semimartingale Markov processes in terms of Wiener processes and Poisson random measures, Seminar on Stochastic Processes (1981), Birkhauser: Birkhauser Boston) · Zbl 0531.60068 |
| [28] | Dipietro, M., Bayesian Inference for Discretely Sampled Diffusion Processes with Financial Applications (2001), Carnegie Mellon University, (Ph.D. thesis) |
| [29] | Duffie, D.; Gârleanu, N., Risk and valuation of collateralized debt obligations, Financ. Anal. J., 57, 41-59 (2001) |
| [30] | Duffie, D.; Pan, J.; Singleton, K., Transform analysis and asset pricing for affine jump-diffusions, Econometrica, 68, 1343-1376 (2000) · Zbl 1055.91524 |
| [31] | Egorov, A. V.; Li, H.; Xu, Y., Maximum likelihood estimation of time-inhomogeneous diffusions, J. Econometrics, 114, 107-139 (2003) · Zbl 1085.62131 |
| [32] | Errais, E.; Giesecke, K.; Goldberg, L., Affine point processes and portfolio credit risk, SIAM J. Financial Math., 1, 642-665 (2010) · Zbl 1200.91296 |
| [33] | Filipović, D.; Mayerhofer, E.; Schneider, P., Density approximations for multivariate affine jump-diffusion processes, J. Econometrics, 176, 93-111 (2013) · Zbl 1284.62110 |
| [34] | Giesecke, K.; Kakavand, H.; Mousavi, M., Exact simulation of point processes with stochastic intensities, Oper. Res., 59, 1233-1245 (2011) · Zbl 1234.62126 |
| [36] | Giesecke, K.; Smelov, D., Exact sampling of jump diffusions, Oper. Res., 61, 894-907 (2013) · Zbl 1291.60171 |
| [37] | Glasserman, P., Monte Carlo Methods in Financial Engineering (2004), Springer-Verlag: Springer-Verlag New York · Zbl 1038.91045 |
| [38] | Hall, P., The Bootstrap and Edgeworth Expansion (1995), Springer-Verlag: Springer-Verlag New York |
| [39] | Hawkes, A., Spectra of some self-exciting and mutually exciting point processes, Biometrika, 58, 83-90 (1971) · Zbl 0219.60029 |
| [40] | Hayashi, M.; Ishikawa, Y., Composition with distributions of Wiener-Poisson variables and its asymptotic expansion, Math. Nachr., 285, 619-658 (2012) · Zbl 1252.60051 |
| [41] | Ikeda, N.; Watanabe, S., Stochastic Differential Equations and Diffusion Processes (1989), Elsevier: Elsevier North-Holland · Zbl 0684.60040 |
| [42] | Jin, X.; Zhang, A. X., Decomposition of optimal portfolio weight in a jump-diffusion model and its applications, Rev. Financ. Stud., 25, 2877-2919 (2012) |
| [43] | Johannes, M. S.; Eraker, B.; Polson, N., The impact of jumps in volatility and returns, J. Finance, 53, 1269-1300 (2003) |
| [44] | Johnson, N. L.; Kotz, S.; Balakrishnan, N., Continuous Univariate Distributions 1 (1996), Wiley: Wiley New York · Zbl 0811.62001 |
| [45] | Kanwal, R. P., Generalized Functions: Theory and Applications (2004), Birkhäuser: Birkhäuser Boston · Zbl 1069.46001 |
| [46] | Karatzas, I.; Shreve, S. E., Brownian Motion and Stochastic Calculus (1991), Springer-Verlag: Springer-Verlag New York · Zbl 0734.60060 |
| [47] | Karlin, S.; Taylor, H., A Second Course in Stochastic Processes (1981), Academic Press: Academic Press Boston · Zbl 0469.60001 |
| [48] | Kloeden, P. E.; Platen, E., The Numerical Solution of Stochastic Differential Equations (1992), Springer-Verlag: Springer-Verlag Berlin Heidelberg · Zbl 0925.65261 |
| [49] | Kou, S., A jump diffusion model for option pricing, Manage. Sci., 48, 1086-1101 (2002) · Zbl 1216.91039 |
| [50] | Li, C., Maximum-likelihood estimation for diffusion processes via closed-form density expansions, Ann. Statist., 41, 1350-1380 (2013) · Zbl 1273.62196 |
| [52] | Liu, J.; Longstaff, F.; Pan, J., Dynamic asset allocation with event risk, J. Finance, 58, 231-259 (2003) |
| [53] | Merton, R., Option pricing when underlying stock returns are discontinuous, J. Financ. Econ., 3, 125-144 (1976) · Zbl 1131.91344 |
| [54] | Merton, R. C., Theory of rational option pricing, Bell J. Econ., 4, 141-183 (1973) · Zbl 1257.91043 |
| [55] | Mykland, P., Asymptotic expansions and bootstrapping-distributions for dependent variables: a martingale approach, Ann. Statist., 20, 623-654 (1992) · Zbl 0759.62011 |
| [56] | Mykland, P., Asymptotic expansions for martingales, Ann. Probab., 21, 800-818 (1993) · Zbl 0776.60047 |
| [57] | Mykland, P.; Zhang, L., The econometrics of high frequency data, (Kessler, M.; Lindner, A.; Sørensen, M., Statistical Methods for Stochastic Differential Equations (2010), Chapman & Hall/CRC Press: Chapman & Hall/CRC Press New York), 109-190 · Zbl 1375.62023 |
| [58] | Pan, J., The jump-risk premia implicit in options: evidence from an integrated time-series study, J. Financ. Econ., 63, 3-50 (2002) |
| [59] | Pan, J.; Liu, J., Dynamic derivative strategies, J. Financ. Econ., 69, 401-430 (2003) |
| [60] | Protter, P., Stochastic Integration and Differential Equations (1990), Springer-Verlag: Springer-Verlag Berlin Heidelberg · Zbl 0694.60047 |
| [61] | Schaumburg, E., Maximum Likelihood Estimation of Jump Processes with Applications to Finance (2001), Princeton University, (Ph.D. thesis) |
| [62] | Singleton, K. J., Empirical Dynamic Asset Pricing: Model Specification and Econometric Assessment (2006), Princeton University Press: Princeton University Press Princeton · Zbl 1094.91030 |
| [63] | Stramer, O.; Bognar, M.; Schneider, P., Bayesian inference for discretely sampled markov processes with closed-form likelihood expansions, J. Financ. Econ., 8, 450-480 (2010) |
| [64] | Watanabe, S., Analysis of Wiener functionals (Malliavin calculus) and its applications to heat kernels, Ann. Probab., 15, 1-39 (1987) · Zbl 0633.60077 |
| [65] | Xiu, D., Hermite polynomial based expansion of european option prices, J. Econometrics, 179, 158-177 (2014) · Zbl 1298.91171 |
| [66] | Yoshida, N., Asymptotic expansions for statistics related to small diffusions, J. Japan Stat. Soc., 22, 139-159 (1992) · Zbl 0778.62018 |
| [67] | Yu, J., Closed-form likelihood approximation and estimation of jump-diffusions with an application to the realignment risk of the chinese yuan, J. Econometrics, 141, 1245-1280 (2007) · Zbl 1418.62291 |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
